and \(h^*:\mathbb {R}^m\rightarrow \overline{\mathbb {R}}\)  is the conjugate of \(h\) . The transition from the sup-expression to \(h^\nu (x)\)  is achieved by recognizing that the former is the conjugate of the function \(u\mapsto h(u +v ) + {\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert u\Vert _2^2\)  evaluated at \(y^\nu \) . This in turn is expressed by the conjugate of \(h\)  and the conjugate of \({\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert \cdot \Vert _2^2\) ; see [1]} as well as 11(3) and Theorem 11.23(a) in [2]}.
Thus, for any \(y^\nu \in \mathbb {R}^m\) , the approximating problem
\(\mathop {\rm minimize}_{u\in \mathbb {R}^m, x\in \mathbb {R}^n} f_0(x) + h\Big (u + \mathop {\sum }\nolimits _{i=1}^s p_i^\nu G_{i}(x)\Big ) + {\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert u\Vert _2^2 - \langle y^\nu , u\rangle \) 
